A solution to the Cauchy dual subnormality problem for 2-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator T′:=T(T⁎T)−1 of a 2-isometry T is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if T is a 2-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then T′ is subnormal if and only if T satisfies the (unperturbed) kernel condition. The second counterexample arises from a 2-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if T is a 2-isometry satisfying the kernel condition or a quasi-Brownian isometry, then T′ is subnormal. We construct a 2-isometric adjacency operator T of a rooted directed tree such that T does not satisfy the kernel condition, T is not a quasi-Brownian isometry and T′ is subnormal.